stefano zippilli et al- scheme for decoherence control in microwave cavities

8/3/2019 Stefano Zippilli et al- Scheme for decoherence control in microwave cavities

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Scheme for decoherence control in microwave cavities

Stefano Zippilli,1 David Vitali,1 Paolo Tombesi,1 and Jean-Michel Raimond21 Dipartimento di Fisica and Unita INFM, Universita di Camerino, via Madonna delle Carceri, I-62032 Camerino, Italy

2Laboratoire Kastler Brossel, Departement de Physique de lEcole Normale Superieure, 24 rue Lhomond,

F-75231 Paris Cedex 05, France

Received 18 November 2002; published 1 May 2003

We present a scheme that is able to protect the quantum states of a cavity mode against the decoheringeffects of photon loss. The scheme preserves quantum states with a definite parity, and improves previous

proposals for decoherence control in cavities. It is implemented by sending single atoms, one by one, through

the cavity. The atomic state gets first correlated to the photon number parity. The wrong parity results in an

atom in the upper state. The atom in this state is then used to inject a photon in the mode via adiabatic transfer,

correcting the field parity. By solving numerically the exact master equation of the system, we show that the

protection of simple quantum states could be experimentally demonstrated using presently available experi-

mental apparatus.

DOI: 10.1103/PhysRevA.67.052101 PACS numbers: 03.65.Yz, 03.67.a, 42.50.Dv, 42.50.Ar

I. INTRODUCTION

In recent years, considerable effort has been devoted todesigning strategies that is able to counteract the undesired

effects of the coupling with an external environment. No-

table examples are quantum-error-correction codes 1 anderror-avoiding codes 2, both based on encoding the state tobe protected into carefully selected subspaces of the jointHilbert space of the system and a number of ancillary sys-tems. The main limitation for the efficient implementation ofthese encoding strategies for combating decoherence is thelarge amount of extra resources required 3. Correcting allpossible one-qubit errors requires at least five qubits 4. Thisnumber rapidly increases if fault tolerant error correction isalso considered. For this reason, other alternative approaches

that do not require ancillary resources have been pursued anddeveloped in parallel with encoding strategies. These deco-herence control schemes may be divided into two main cat-egories: open-loop 510 and closed-loop or quantum feed-back 11 strategies 1215.

In open-loop techniques also called dynamical decou-pling schemes, the system is subject to an external, suitablytailored, time-dependent driving. The external control Hamil-tonian is chosen on the basis of a limited, a priori, knowl-edge of the system-environment dynamics, in order to realizean effective dynamical decoupling of the system from theenvironment. The main idea behind these open-loop schemesoriginates in refocusing techniques in NMR spectroscopy16

, but they have been recently transposed in many differ-ent contexts, such as the inhibition of the decay of an un-

stable atomic state 8, the suppression of magnetic state de-coherence 9, and the reduction of heating effects in linearion traps 10. The main drawback of open-loop decouplingprocedures is that the timing constraints are, particularly,stringent. In fact, the decoupling interaction has to be turnedon and off at extremely short time scales, even faster than thetypical environmental time scale 5,7,10. An approach toquantum state protection related to decoupling schemes isrepresented by reservoir engineering schemes 1719, inwhich an external driving is used to create an effective res-

ervoir for the system. In such a way, the state to be protected

becomes a stationary state of the modified dynamics. Ex-

amples have been proposed for the center-of-mass motion oftrapped ions 17,19 and for atomic internal states 18.

It is interesting to notice that quantum-error-correction

codes, error-avoiding codes, and decoupling schemes can bedescribed in a unified framework based on the representa-tions of the algebra of errors the algebra generated by the setof system operators describing the effects of the environ-ment 20,21. In the error-algebra framework, quantum in-formation is protected using symmetry. In the case ofdecoherence-free subspaces, the symmetry naturally exists inthe interaction with the environment. In the case of decou-pling techniques, symmetry is induced by the added drivingHamiltonian. Finally, in quantum-error-correction codes,

symmetry exists implicitly within the larger Hilbert space ofthe system and ancillas 20.

Closed loop techniques represented the first attempt tocontrol decoherence 12,13. In this case, the system to beprotected is subject to appropriate measurements, and theclassical information obtained from this measurement is usedfor real-time correction of the system dynamics see, for ex-ample, Ref. 22 for a recent experimental demonstration ofstate stabilization by feedback. This technique shares, there-fore, some similarities with quantum-error correction, whichalso checks which error has taken place and eventually cor-rects it. However, the main limiting aspect of feedbackschemes is the need of a measurement, which is always in-

evitably prone to errors and finite detection efficiency. Forthis reason, recent attempts have tried to improve theseclosed-loop schemes by avoiding the explicit measurementstep, resorting to what may be called fully quantum feed-back schemes, in which sensors, controller, and actuatorsare themselves quantum systems and interact coherently withthe system to be controlled 23. In this case, the entire feed-back loop is coherent and is not limited by measurementinefficiencies. In fact, some of us have already proposed ascheme of this kind 15. It improved an existing closed-loopscheme for decoherence control 14, by replacing the mea-surement step an atomic detection with the coherent inter-

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action with a quantum controller, represented by a high-Q

cavity. The quantum coherent interaction allows an auto-

matic correction of the dynamics, without requiring an ex-

plicit measurement, similar to what happens in quantum er-

ror correction codes.

In this paper, we proceed further along this direction by

introducing a significant simplification of the automatic

scheme of Ref. 15. As in Ref. 15, the present scheme is

designed to protect an arbitrary quantum state of a micro-

wave cavity mode with a given parity against the decohering

effects of photon loss. However, in the present proposal, the

whole feedback loop is realized by a single atom crossing the

cavity. The atom first measures the parity of the field and

then performs, in the last stage of the cavity crossing, the

state correction; whenever it is needed. For this reason, this

scheme is another example of a fully quantum feedback loop

23, which employs very limited resources, namely, a single

atom playing the roles of sensor, controller, and actuator. In

the scheme of Ref. 15 instead, the first atom is the sensor

and the second high-Q cavity is the controller. The second

atom is used as the actuator. The proposed strategy sharesalso some analogies with quantum-error-correction codes.

The error to be corrected is the loss of one photon, which

drives the state out of a parity eigenspace. With this respect,

the two-level atom implementing the scheme plays a role of

the error syndrome, because its state denotes the eventual

presence of an error, that is, of a wrong parity.

The simplifications brought by the present scheme are rel-

evant since they make the present proposal much easier to

implement than that of Ref. 15, because it does not need a

second high-Q cavity and the use of a second atomic source.An easy implementation is an important asset. In spite ofseveral theoretical proposals for decoherence control, there

have been only few experimental demonstrations. A simpleexample of decoherence-free subspace immune frommagnetic-field noise has been demonstrated with two trappedions in Ref. 24, while error-correction codes for single-qubit errors has been demonstrated only in NMR quantuminformation processors 25. Outside the usual application inNMR refocusing techniques, dynamical decoupling schemeshave been implemented only in Ref. 26, where a proof-of-principle demonstration for a photon polarization qubit withartificially added decoherence has been realized see alsoRef. 27 for a recent demonstration of encoded decouplingschemes in NMR systems. A simple demonstration of fullyquantum feedback has been given in the case of a three-nuclear spin system in Ref. 28, but no closed-loop decoher-ence control scheme has been demonstrated yet. The even-tual implementation of the present proposal is important alsobecause it would represent the first demonstration of the con-trol of an intrinsic and unavoidable decoherence source, pho-ton loss, instead of the control of an added noise.

The paper is organized as follows: In Sec. II, the cavityQED model under study is described, and the quantum stateprotection scheme is presented in detail. In Sec. III, the per-formance of the proposed scheme is studied by solving nu-merically the dynamical evolution of the system. Differentexamples of initial quantum states of the radiation mode to

protect will be considered. Section IV is for concluding re-marks.

II. THE STATE PROTECTION SCHEME

The general purpose of decoherence control schemes is toprotect a given subspace of a system Hilbert space, andquantum coherent evolutions within it. The conditions under

which these noiseless subsystems exist, and that universalquantum computation within them is possible have been al-ready illustrated in the recent literature, especially in the caseof qubits 5,20,21,29,30. The particular case of infinite-dimensional systems such as radiation modes is of funda-mental importance for quantum communication schemes. Inthis case, the extension of the quantum-error-correctioncodes for qubits 1,4 to the infinite-dimensional case hasbeen shown in Ref. 31, while a general and an elegantapproach to the construction of error-correction codes in con-tinuous variable systems has been proposed in Ref. 32.However, the experimental realization of these schemes isdifficult because the error-correction codes of Ref. 31 re-

quire coupling efficiently a large number of modes, while theproposal of Ref. 32 employs linear superpositions ofsqueezed states as encoded states, which are very difficult togenerate.

Here, we shall focus on the case of a radiation modeconfined in a cavity. The first examples of quantum gates andquantum state manipulations have been demonstrated in thiscontext 3338. Moreover, cavity modes could represent thenodes of a quantum network of multiple atom-cavity systemslinked by optical interconnects 39.

In electromagnetic cavities, decoherence is mainly of dis-sipative origin and it is associated with the photon losses dueto diffraction and to the transmission and absorption of themirrors. In the general case where the reservoir of the con-tinuum of electromagnetic modes is at thermal equilibrium attemperature T, the dynamics is well described by the masterequation in the frame rotating at the mode frequency )40

L a

2N1 2aa aaa a

2N 2a aaa aa , 1

where is the field density matrix and a is the annihilationoperator of the cavity mode, is the cavity decay rate, andN exp(/kT)11 is the equilibrium thermal photonnumber. The cavity mode is affected by two kinds of errors,photon loss with rate (N1) ] and thermal photon creationwith rate N). However, in many cases, one has N1.Photon loss is then, by far, the predominant source of deco-herence.

In Ref. 14, a closed-loop scheme for protecting a ge-neric state of a cavity mode has been proposed, based on thesimple idea of giving back the photon as soon as it is lost. Inthe case where the sensor is represented by a single-photonphotodetector with quantum efficiency continuously moni-

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toring the cavity, the dynamics in the presence of feedback is

described by the master equation in the case N0) 14

1

2 2aaa aa a

2 a a,aa ,. 2

In the case of perfect detection (1), cavity damping is

therefore replaced by an unconventional phase-diffusion pro-

cess. In the ideal case, the only well-preserved states are the

Fock states. However, since the phase-diffusion process is

very slow, the resulting quantum state protection is still sig-

nificant for other states 14.

In the case of microwave cavities, there are no efficient

single-photon detectors besides atoms crossing the cavity

mode. The photon counting can be replaced in this case by a

field parity measurement 14,4144. This measurement canbe efficiently performed by using a dispersive atom-field in-

teraction revealed by a Ramsey interferometry setup 45. If

these parity measurements are repeated at short-time inter-

vals, so that multiple photon losses between them are negli-

gible, they reveal unambiguously the photon losses and re-

place a single-photon photodetector. This is precisely the

stroboscopic measurement scheme proposed in Ref. 14 for

the cavity QED microwave experiments described in details

in Ref. 46. The price to pay when using parity measure-ments instead of photon counting is that only states with agiven parity can be protected.

Reference 15 improved this closed-loop decoherence

control scheme by transforming it into one of the first ex-amples of fully quantum feedback loop. In Ref. 14, thefeedback loop involved a first atom probing the parity of thecavity mode. The final state of the first atom, correlated tothe field parity, was measured by a state-selective atomicdetector. Depending upon the result of this measurement, adedicated electronics could send a second atom through thecavity. This atom would emit a photon in the mode, correct-ing the effect of photon loss and restoring the initial fieldparity. Therefore, in this case, both the sensor first atomdetector and the controller the electronics are essentiallyclassical, while only the actuator the second atom is aquantum system. In Ref. 15, the detector and the control-ling electronics are replaced by a second high-Q microwavecavity, resonantly interacting with the two atoms. This cavitybecomes the controller and the feedback loop has becomecompletely quantum. Here, we propose a further improve-ment of this protection scheme, making its experimentalimplementation easier. The present scheme is based again onthe measurement of the cavity mode parity, but it involvesonly one atom, which, in passing through the cavity, firstmeasures and then corrects the state of the mode whenneeded. The simplification of the design is evident, with asingle atom realizing the whole loop, by playing all the rolesof sensor, controller, and actuator.

The physical system and the protection scheme in detailThe microwave cavity QED setup that we have specifi-

cally considered for the implementation of the proposed de-coherence control scheme is described in detail in Ref. 46,in which either generation of nonclassical states of the radia-tion 35,36, and coherent quantum state manipulation 38have been already demonstrated.

A sketch of the setup is shown in Fig. 1. Its central part isa superconducting cavity C in a Fabry-Perot configuration,cooled down at about 1 K. It sustains two Gaussian fieldmodes with the same spatial structure and orthogonal linearpolarizations. They are separated by a frequency interval of128 kHz around 51.1 GHz. The cavity modes can be

driven by a tunable classical source S. These two modes caninteract in a controlled way with single, velocity-selectedatoms effusing from oven O. The atoms are prepared one ata time in long-lived lifetime 30 ms), circular Rydbergstates, e principal quantum number 51 and g principalquantum number 50, in box B. The atoms then interact withthe cavity, quasi resonant on the eg transition. The atom-field coupling is measured by the single-photon Rabi fre-quency, which is time dependent because of the mode Gauss-ian spatial structure. At time t, the Rabi frequency writes(t)0expv

2t2/w2, where w6 mm is the modewaist, 0/224.5 kHz 47, and t0 corresponds to theatom crossing cavity axis. The detuning from the cavitymode resonance frequency, ( t)

eg( t), can be

changed in time in a controlled way using the Stark shiftinduced by a uniform electric field applied across the cavitymirrors. The two-level atom can be manipulated also throughmicrowave pulses generated by the tunable source S in alow-Q transverse mode. The final atomic state is recorded bythe state-selective detector D.

The whole protection process is realized by the atom dur-ing its transit through the cavity mode. The first part of theinteraction time is used for the measurement of the cavitymode parity, while the second part is used for the possiblestate correction. A long interaction time is therefore needed,

FIG. 1. Scheme of the experimental apparatus. The Rb atomic

beam effuses from oven O and circular Rydberg atoms are prepared

one at a time in box B. They cross the high-Q microwave cavity C

whose state we want to protect, and which can be driven by the

source S. The classical source S is used for atomic state manipu-

lations, and D is the field-ionization detector.

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requiring atoms with a moderate velocity. We are consideringhere velocities around v100 m/s, which are straightfor-wardly obtained in the experiments without any need ofatomic beam cooling 44. The feedback atoms, sent one byone, are initially prepared in the excited state e . The feed-back atoms are finally detected in the field-ionization detec-tor D. In a single experimental sequence, we might thus ac-cess the individual quantum trajectory of the cavity field,conditioned to the atomic detections. We are, however, inter-ested here in an unconditional decoherence control scheme,able to preserve any quantum state with a given parity. Forthis reason, we assume that the information about the indi-

vidual atomic state is finally discarded in the data analysis,and we thus consider only quantum averages of several in-dividual trajectories. Note that keeping the atomic state in-formation and accessing to individual trajectories leads to adifferent, conditional, protection scheme, which will be dis-cussed elsewhere. The state of the system just before a ge-neric atom enters the cavity is thus e e, where is thereduced state of the cavity mode. We consider only the cavitymode to protect, while the other quasiresonant mode is sim-ply a spectator mode, even though it has been taken intoaccount in the numerical simulations described in the follow-ing section.

The parity measurement 44 is performed using a Ram-sey interferometry scheme 45, involving a dispersive inter-action in which the eg transition is light shifted by thecavity mode 33, sandwiched between two /2 pulses mix-ing e and g before and after the dispersive interaction. Thetwo /2 pulses are generated by the source S in a low-Qtransverse mode in the cavity structure 46. In order to mini-mize a spurious coupling of the Ramsey source S with thesuperconducting cavity modes, the atomic transition isshifted far away from the cavity resonance by the Stark ef-fect at the time of the /2 pulses see Fig. 2 showing thespatial dependence of the atomic detuning within the cavity,providing a schematic description of the protection scheme.

A proper pulse shape tailoring is used to decrease even fur-ther this spurious coupling 44.

The dispersive interaction between the atom and the cav-ity mode is obtained for sufficiently large atomic detuning,i.e., when (t)/(t)1, and for adiabatic variations of theparameters. The corresponding Hamiltonian in the frame ro-tating at the cavity mode frequency is then

Hdisp t

2 e egg

2 t

t gga ae eaa . 3

The associated unitary evolution is given by

Udisp,e eei/2eia

agge i/2e ia

a,4

where dt( t)2( t)/(t) and dt2( t)/( t).As shown in Refs. 14,15,42,43, a conditional phase shiftper photon equal to is needed for a parity measurement,which implies adjusting the detuning and the duration of thedispersive interaction in such a way that /2.

The first stage of the feedback loop, describing the paritymeasurement, is given therefore by the transformation

eeU/2Udisp, 2 U/2e

eU/2

Udisp, 2

U/2 , 5

where U/2 ( eg)e( ge)g/2. Using thisfact, the state of the system after step c of Fig. 2 can be

rewritten as

eiaa/2 1e ie ia

a

2 e 1e ie ia

a

2 g

e 1eieiaa

2

g 1eieiaa

2 e ia a/2. 6

It is evident that the measurement of the cavity mode parityis obtained if the phase is appropriately adjusted so thate i1. Each atomic state is then unambiguously corre-lated with a parity eigenvalue. The phase can be adjustedto any desired value, by strongly detuning the atom from thecavity with a very short Stark-shift pulse, which has a neg-ligible effect on step d of Fig. 2. In the second part ofthe interaction time, the atom is used to deliver a photon tothe cavity when the wrong parity has been measured. Theexcited state e has thus to be correlated with the wrongparity component. By choosing 0 or , we canchoose which kind of parity eigenstates of the cavity mode isprotected against photon losses.

FIG. 2. Variation of the atomic detuning eg thick line

as a function of the position within the cavity C. The two horizontal

lines H and L denote the frequencies of the two cavity modes sepa-

rated by 128 kHz. The two vertical dotted lines at 6 mm

denote the cavity waist. The variation of the detuning is obtained by

means of the Stark shift induced by an electric field applied through

the cavity mirrors. The various steps of the protection scheme are

visible. The two /2 Ramsey pulses a and c, the dispersive

-phase shift b, and the phase-tuning Stark-shift pulse d repre-

sent the parity measurement stage of the scheme. The pulse g

i of step e and the adiabatic transfer step f constitute instead

the correction stage see the text for details.


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To be specific, we consider from now on the protection ofodd cavity states. This implies choosing e i1, so thate is associated with even states. Finally, note that the over-all /2 phase-space rotation of the cavity mode in Eq. 6can be eliminated by simply adjusting the phase of the ref-erence field in S, so that the state of the system at the end ofthe measurement stage can be written as

P even eP od dg]ePevengPod d, 7

where

P even1e ia

a

2, 8

P od d1e ia

a

29

are the projectors onto the even and odd parity eigenspaces,respectively.

In the second part of its interaction time with the cavity,

the atom corrects the cavity state component with the wrongparity, by transferring its excitation to it. As in Ref. 15, thisis done using adiabatic transfer. When the atomic detuning isadiabatically changed from a large positive to a large nega-tive value, the system remains in the instantaneous dressedstate see Ref. 15, realizing therefore the transformation

e,ng ,n1 n. 10

The photon emission is thus independent of the cavity modestate, an essential feature for state-independent protection.

If the atom is in state g, corresponding to a field measuredto be in the right parity state, the opposite adiabatic trans-fer g ,ne,n1 could take place, resulting in a spoiledparity. We have thus to get rid of the atom when it exits theparity measurement in state g. This can be achieved by tun-ing the classical source S on resonance with the gi tran-sition (i is a lower circular Rydberg state with principalquantum number 49 46 and realizing a pulse gi. Theatom is then shelved in state i, which does not interactwith the cavity mode. This prevents this unwanted transfer tooccur.

The ideal adiabatic transfer can be formally described bythe operator

Uadiagea

1

aaega

1

a aii, 11

so that the state of the atom-cavity mode system at the end ofthe atomic passage is

ga 1aa P eveniP od d P even1

aa agP od di .

12

Therefore, in each cycle, the cavity mode is either projectedinto the correct odd parity eigenspace, or is corrected viaadiabatic transfer when it has a wrong even parity. The two

possibilities could be distinguished by detecting the exitingatoms, respectively, in i or in g, selecting in this way one ofthe quantum trajectories of the cavity mode conditional state.However, a fully quantum feedback has not to rely on theclassical information provided by the atomic state detection,and it is necessarily unconditional. Therefore, we discard theinformation about the individual atomic state, and by tracingover the atom, we get that a generic feedback cycle, i.e., a

complete atomic passage, can be described by the followingmap for the cavity mode state :

a1

aaP evenP even

1

aaaP od dP od d . 13

These are the unitary manipulations characterizing the feed-back scheme. However, in practice, these manipulations actsimultaneously with the decohering effect of the thermal en-vironment described by master equation 1, and which areresponsible for the errors single-photon losses that thescheme is designed to correct for. The resulting evolution isno more unitary and described by the simple map of Eq.

13. The photon losses contaminate the scheme, and theprojections onto the parity eigenspaces will be no moreexact.

III. NUMERICAL RESULTS

The performance of the proposed protection scheme un-der realistic conditions has been studied by solving numeri-cally the master equation describing the dynamics of thewhole system, composed by the two nondegenerate high-Qcavity modes and the two-level atom crossing them. We haveincluded also the higher-frequency mode with frequency Hand annihilation operator aH), even though we are not inter-

ested in its state. It is supposed to play only the role of aspectator in the process. However, in the apparatus describedin Ref. 46, its frequency is very close to that of the usefulmode. It could be a source of imperfections in the feedbackscheme by producing uncontrolled phase shifts on the atomduring the parity measurement.

We have considered the following master equation for thedensity operator of the whole system T:

Ti

H,TL a TL aHT , 14

where the superoperator L(a) has been defined in Eq. 1,

and

HaH

aH t

2 e eggi t aeg

a g ei t aHegaH g e 15

is the total Hamiltonian of the system in the frame rotating atthe frequency of the mode of interest. We assume that theatoms are sent one by one, with a spatial separation of 40mm. Since the cavity diameter is 50 mm and the mode waistis 6 mm, this guarantees that the atoms interact with the


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cavity mode one at a time, so that two-atoms effects are

avoided. Each feedback cycle lasts exactly the time the atom

takes to cross the cavity region of length 40 mm around the

cavity axis see Fig. 2. Every cycle immediately follows the

preceding one, starting with the atom entering the interaction

region just when the preceding one has left it. The feedbackatom is always initially prepared in state e so that, at thebeginning of each cycle, the state of the whole system is

e eou t, where ou t is the reduced state of the two modesat the end of the preceding cycle.

The master equation has been solved in a truncated Fockbasis for both modes, using the parameter values of the ex-perimental apparatus described in Ref. 46 see the preced-ing section. We have also assumed that both cavity modesare coupled to a thermal reservoir with a mean photon num-ber N0.8. This means that thermal excitation from the res-ervoir is not negligible. One might thus expect that the pro-posed state protection scheme, designed to correct for photonlosses only, may not work properly in this case. We shall seethat this is not the case because photon loss is still more thantwice more probable than thermal excitation. This is enough

for our feedback scheme to achieve a significant state pro-tection.

As it has been already discussed in Refs. 14,15, a crucialparameter is the ratio between the time duration of the feed-back cycle coinciding with the interaction time of the atomand the relaxation time of the cavity mode 1Tr . It isevident that this ratio has to be as small as possible. Fastatoms would be preferred. However, the feedback cycle isoptimal at moderate velocities, since all the atomic manipu-lations have to fit within the cavity crossing time. Note thatthe dispersive step of the parity measurement critically de-pends on the interaction time. In order to fulfill the-phase-shift condition dt2(t)/(t)/2 , the

faster the atom, the smaller the detuning . However, forsmall values of, the dispersive Hamiltonian Eq. 3 is nolonger a good approximation of the total Hamiltonian of Eq.15. The correlation between the atomic state and the cavitymode parity of Eq. 7 is thus imperfect. We have seen thatthe best protection results are obtained with atomic velocitieswithin the range of 80110 m/s. A clear example of quantumstate protection in the case of an initial odd superposition oftwo coherent states with opposite phases Schrodinger catstate, 1 with 1.8, is described in Fig. 3,where snapshots of the time-evolved Wigner function, bothin the presence top and in the absence bottom of protec-tion, are presented. Note that a proper experimental check ofthe feedback procedure would be to map out the cavity stateWigner function, using the technique demonstrated in Ref.44.

Atom No. 1 refers to the first atom generating the cat stateusing the scheme already described in Ref. 42, and em-ployed in the cat state experiment of Ref. 35. This meansthat, in order to be more realistic, we have always consideredthe protection of the effective quantum state generated by thescheme, and not of an initial, ideal, quantum state. In thepeculiar case of the Schrodinger cat state of Fig. 3, the gen-eration is obtained assuming the initial coherent state injected in the cavity by the source S, and applying just the

parity measurement described in the preceding section. The

odd cat state is generated by postselection, when the atom isdetected in state g 42. The elapsed time is measured alsoin the case with no protection, where no atom is used interms of the number of crossing atoms n: the time elapsedfrom the exit of the first atom generating the state out of theinteraction region is tnL(n1)/v, where L40 mm. Fig-ure 3 refers to an atomic velocity v80 m/s and a cavity

mode relaxation time Tr10 ms, which therefore corre-sponds to 20 atomic passages. The comparison with the un-protected evolution indicates a good quantum state protec-tion until the ninth atom, even though the decoherence timein this case is tde c(2

2)11.54 ms 48, correspond-ing to three atomic passages. The detuning used in the dis-persive stage step b of Fig. 2 is /2197 kHz, whilewe have used a parabolic variation of the detuning ( t)around resonance for the adiabatic photon transfer step fof Fig. 2 because it turned out to be the most effective one.Both the /2 Ramsey pulses steps a and c, and the pulse of step e from the source S had a duration of1.25 s, with their intensity and frequency consistently

tuned. The Stark-shift pulse of step d, needed to tune thephase shift so that e i1, lasted for 1.25 s, with adetuning of about 1 MHz.

A more quantitative description of the capabilities of theprotection scheme is given by Fig. 4, where the time evolu-tion of the fidelity F(t)1(t)1 and the parity P(t) n(1)

nnn ( t) for the odd cat state of Fig. 3 are shown at

a fixed atom velocity v80 m/s, and at two different valuesof the cavity mode relaxation time Tr1,10 ms. The starsconnected by the dotted line refer to the protected evolution,while the diamonds linked by the full line refer to the evo-lution with no protection. The fidelity is appreciably im-proved when Tr10 ms, and a small improvement can be

seen even when Tr1 ms. This is not surprising, because asingle atomic passage lasts for 0.5 ms, which is equal to halfrelaxation time in this latter case. On the other hand, we cansee that the proposed protection scheme is actually a verygood parity preservation scheme. In fact, the parity of theinitially generated state atom No. 1 is satisfactorily pre-served in time, even in the case of Tr1 ms. In such a case,the initial state is far from being an ideal odd cat state be-cause, due to photon losses, the projection onto the oddeigenspace of Eq. 9 is far from being effectively realized.With this respect, our scheme is not fault tolerant, i.e., it doesnot work perfectly in the presence of losses. Not only theprojections onto the parity eigenspaces, but also the adiabatictransfer in the correction stage is not perfect under realisticconditions. Its efficiency when Tr1 ms is about 90%, andit is due not only to photon losses, but also to the fact that theinitial positive value of the detuning cannot be taken as largeas required, because the atom would come close to resonancewith the high-frequency mode and transfer its excitation to itrather than to the mode to be protected.

The most recent experiments with the cavity QED appa-ratus described in Ref. 46 have been performed with a mi-crowave mode with a relaxation time Tr1 ms. We have,however, considered also longer relaxation times becausethere are realistic prospects to achieve somewhat longer cav-


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ity damping times soon. We have also investigated the per-formance of the scheme in the case of higher atomic veloci-ties. Protection of the odd cat state remains essentiallyunchanged up to v110 m/s, and clearly worsens for veloci-

ties larger than 150 m/s. For such velocities the dispersive-phase shift can be no more realized in a satisfactory way.The scheme is suitable to protect any quantum coherent

superposition with a given parity, and not only cat states. Wehave in fact also considered the case of an initially generatedsuperposition of two Fock states 2(13)/2. Thetime evolution of the reduced cavity mode density matrix inthe Fock basis, in the presence of the protection scheme, iscompared with that with no protection in Fig. 5, considering,as in Fig. 3, Tr10 ms and v80 m/s. The other parametervalues of the scheme are the same as those used in the catstate case of Fig. 3.

We have considered also in this case, the protection of an

initial state effectively generated within the apparatus by the

first atom. The generation of the superposition state 2(13)/2 can be achieved in the following way. Ini-

tially, the atom is injected in state e with the cavity mode inthe vacuum state the thermal cavity field can be erased by

sending through it a train of absorbing atoms 46. Then, a

photon pump mechanism 49 can be used to transfer pho-

tons into the cavity mode. The atomic excitation is first trans-

ferred to the mode via a resonant atom-cavity interaction.

The atom is then reset to the excited state leaving the cavity

undisturbed by simultaneously Stark-shifting the atomic

levels well out of resonance from the cavity mode, and ap-

plying a pulse on the transition eg . By repeating thissequence, one can generate an arbitary Fock state n . In the

FIG. 3. Time evolution of the

Wigner function of the cavity

mode for an initial odd Schro-

dinger cat state, 1, with 1.8, with topand without bottom state protec-

tion. The atom velocity is v

80 m/s and the cavity relaxation

time is Tr10 ms, while the other

parameters are given in the text.The elapsed time is measured in

both cases in terms of the number

of atomic passages. The time ori-

gin is given by the exit of the first

atom generating the cat state out

of the interaction region, and then

each atomic passage lasts for

500 s.


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specific case of the state 2 , the generation sequence is asfollows: i resonant pulse yielding e,0g ,1; ii Starkshift and classical pulse giving g ,1e ,1; iii classical /2 pulse on the ei transition yielding e ,1( e ,1i,1)/2; iv resonant interaction with the cavity, realiz-ing the pulse e ,1g,2 , while leaving the i compo-nent untouched; v Stark shift and classical pulse giving

( g ,2i ,1)/2( e,2i,1)/2; vi resonant interac-tion with the cavity mode giving ( e,2i,1)/2(g ,3

i,1)/2; vii classical /2 pulse on the gi transitionyielding the state g( 31)i( 31)]/2, andconsequent atomic detection in state i , generating the de-sired superposition 2. Note that in this case, the genera-tion can be performed in a very short time and in our simu-lation, we have used a velocity v400 m/s for thegenerating atom.

The density matrix of the protected state after 19 atomicpassages see Fig. 5 clearly shows the effect of thefeedback-induced square root of phase diffusion discussedabove. This phase diffusion manifests itself at long times,eventually driving the cavity mode into a stationary statisti-cal mixture of Fock states, corresponding to a rotationallyinvariant Wigner function in phase space 14 see also theWigner function of the protected state after 9 atoms in Fig. 3,where the two Gaussian peaks start to be stretched by phasediffusion.

We have studied the time evolution of the fidelity F( t)2( t) 2 and of the parity in this case, both in thepresence stars connected with a dotted line and in the ab-sence diamonds connected by a full line of quantum feed-backsee Fig. 6. The comparison between the protected andthe unprotected evolution gives results slightly better thanthose of the cat state Fig. 4, because the state especially itsparity is preserved for a longer time up to 20 atoms when

Tr10 ms). As we have seen in Sec. II, and it is discussed

in Refs. 14,15, this protection scheme is a stroboscopic

version of the continuous photodetection feedback scheme

described by master equation 2. Therefore, we expect that

the Fock number states of the cavity mode are particularly

well protected because they are unaffected by the feedback-

induced phase-diffusion process. We have verified this fact in

the case of the one-photon Fock state and the results are

shown in Fig. 7, where the time evolution of the fidelity

F(t)11(t) and of the parity P(t)n(1)nnn (t) are

shown. The first atom is very fast and generates the n1Fock state with a simple resonant interaction e,0g ,1.Figure 7 refers to a cavity relaxation time Tr1 ms, and the

appreciable improvement of the fidelity shows that one can

demonstrate a significant protection of a Fock state using the

presently available experimental apparatus. The n1 stateis easier to protect not only because it is not affected by

phase diffusion, but also because is less sensitive to the dis-

persive step b of the scheme. For this reason, we can use

faster atoms and accordingly smaller values of the detuning

of step b. In Fig. 7, we have used v150 m/s with /2

109 kHz) and even v200 m/s with /2

73 kHz).Finally, it is also interesting to note that the proposed

protection scheme could be even used for the generation of

the n1 Fock state. In fact, as seen in Sec. II, state pro-tection is obtained by projecting and eventually restoring thecomponent with the desired parity. If one starts from a ther-mal equilibrium state with a low mean photon number N sothat nn are very small for n2, the successive applicationof the protection scheme for odd states will filter out mainlythe n1 component, which is obviously a stationary stateof the feedback process 14.

FIG. 4. Time evolution of the fidelity and theparity for the initial odd Schrodinger cat state of

Fig. 3. The stars connected by dotted line refer to

the protected evolution, while the diamonds

linked by the full line refer to the evolution with

no atomic crossing. Parameter values are the

same as in Fig. 3 except that we have considered

two values of the relaxation time Tr1,10 ms.


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The generation of the one-photon Fock state starting froma thermal equilibrium state with initial cavity mean photonnumber N0.8 is shown in Fig. 8, where, again, the timeevolution of the fidelity F(t)11( t) and of the parity areshown symbols are the same as those of Figs. 4, 6, and 7.In this case, there is no generation step and the first atom isused for feedback already. The plots referring to the casewith no protection are obviously flat because the cavity modeis already at thermal equilibrium. As in Fig. 7, we have onlyconsidered the case with Tr1 ms, and therefore the signifi-cant difference between the results with and without protec-tion in Fig. 8 shows that the Fock state filtering from aninitial thermal distribution could be implemented using theavailable experimental apparatus. Note also that the statesinvolved in the protection scheme are rotationally invariantin phase space, i.e., are diagonal mixtures in the Fock basis,which are less sensitive to the details of the dispersive step

b. Therefore, we have again used fast feedback atoms andconsequently smaller values for the detuning of the disper-sive step. We have used the same values of the single-photonstate of Fig. 7, v150 m/s with /2109 kHz), andv200 m/s with /273 kHz).

IV. CONCLUSIONS

We have presented a scheme for the protection of a ge-neric quantum state of a cavity mode with a definite parity,against the decohering effects of photon losses. The schemeis a further improvement of the quantum feedback schemesdescribed in Refs. 14,15 and is an example of a fullyquantum feedback loop, where sensor, controller, and actua-tor are all quantum systems 23,28. In the present scheme,all these roles are played by a single atom crossing the cavitymode.

FIG. 5. Time evolution of the density matrix in the Fock basis for the initial superposition of Fock states 2(13)/2 with topand without bottom state protection. The atom velocity is v80 m/s, and the cavity relaxation time is Tr10 ms, while the other

parameters are given in the text. The elapsed time is measured in both cases in terms of the number of atomic passages. The time origin is

given by the exit of the first atom generating the state out of the interaction region, and then each atomic passage lasts for 500 s.


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10/12

The scheme presents many analogies with quantum errorcorrection codes, even though in our case, there is no explicitstate encoding. In fact, the state to be corrected is alreadyencoded within a parity eigenspace and the error a single-photon loss maps this state into an orthogonal subspace, thatis, that with opposite parity. Then one corrects the error onlywithin this orthogonal subspace. In this respect, the atomplays the role of the error syndrome indicating the pres-ence of the error. Finally, the correction is automaticallyimplemented only if needed, by means of a kind of

controlled-NOT gate between the atom control qubit and thecavity mode target qubit represented by the two parityeigenspaces.

We have studied the performance of the proposed feed-back scheme in the case of the cavity QED setup describedin Ref. 46. We have numerically solved the exact masterequation by choosing parameter values corresponding tothose of Ref. 46. The only simplification adopted is that wehave assumed that the circular Rydberg atoms are preparedone at a time with probability 1 in the setup. In other words,

FIG. 6. Time evolution of the fidelity and theparity for the initial superposition of Fock states

of Fig. 5. The stars connected by the dotted line

refer to the protected evolution, while the dia-

monds linked by the full line refer to the evolu-

tion with no atomic crossing. Parameter values

are the same as in Fig. 5, except that we have

considered two values of the relaxation time Tr1,10 ms.

FIG. 7. Time evolution of the fidelity and the

parity in the case of an initial one-photon Fock

state generated by the first atom. The stars con-

nected by the dotted line refer to the protected

evolution, while the diamonds linked by the full

line refer to the evolution with no atomic cross-ing. The cavity relaxation time is Tr1 ms,

while we have considered two velocities, v

150 m/s top and v200 m/s bottom. The

other parameter values are in the text.


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we have assumed a deterministic atomic gun. This condi-tion is not verified by the present setup, where atomic pulseswith a mean number of 0.2 circular atoms per pulse are pre-pared. This fact would lower the efficiency of our scheme.However, this problem could be circumvented by detectingall the feedback atoms exiting the cavity and postselectingonly the events with no missing feedback atom accepting allpossible atomic states.

We have considered well-separated atoms 40 mm be-

tween two successive feedback cycle, just to be sure to avoidany two-atom effect. One could increase the efficiency of theprotection scheme by taking closer feedback atoms. For ex-ample, if we took an atomic separation of 20 mm, the effec-tive feedback cycle time would be halved, and it could stillbe possible to keep two-atoms effects negligible.

The experimental implementation of the present scheme,

especially in the simplest cases of a one-photon Fock state

Figs. 7 and 8 or a superposition of Fock states Figs. 5 and

6 is feasible with the presently available apparatus or with

an apparatus with realistic improvements. Its demonstration

would represent the first implementation of decoherence con-trol schemes based on quantum feedback ideas, and also thefirst example of control of a very common, and almost un-avoidable, source of decoherence such as photon loss.

ACKNOWLEDGMENT

This work was partially supported by the European Unionthrough the IHP program QUEST.

1 P.W. Shor, Phys. Rev. A 52, R2493 1995; A.M. Steane, Proc.

R. Soc. London, Ser. A 452, 2551 1995; E. Knill and R.

Laflamme, Phys. Rev. A 55, 900 1997.

2 L.M. Duan and G.C. Guo, Phys. Rev. Lett. 79, 1953 1997; P.

Zanardi and M. Rasetti, ibid. 79, 3306 1997; D.A. Lidar, I.L.

Chuang, and K.B. Whaley, ibid. 81, 2594 1998.

3 A.M. Steane, Nature London 399, 124 1999.

4 R. Laflamme, C. Miquel, J.-P. Paz, and W.H. Zurek, Phys. Rev.

Lett. 77, 198 1996.

5 L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 1998; L. Viola,

E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 1999; L.

Viola, S. Lloyd, and E. Knill, ibid. 83, 4888 1999; L. Viola

and E. Knill, e-print quant-ph/0208056.

6 M. Ban, J. Mod. Opt. 45, 2513 1998; L.M. Duan and G.C.

Guo, Phys. Lett. A 261, 139 1999.

7 D. Vitali and P. Tombesi, Phys. Rev. A 59, 4178 1999.

8 G.S. Agarwal, Phys. Rev. A 61, 013809 2000; G.S. Agarwal,

M.O. Scully, and H. Walther, Phys. Rev. Lett. 86, 4271 2001;

Phys. Rev. A 63, 044101 2001; J. Gea-Banacloche, J. Mod.

Opt. 48, 927 2001; A.G. Kofman and G. Kurizki, Phys. Rev.

Lett. 87, 270405 2001.

9 C. Search and P.R. Berman, Phys. Rev. Lett. 85, 2272 2000;

Phys. Rev. A 62, 053405 2000.

10 D. Vitali and P. Tombesi, Phys. Rev. A 65, 012305 2002.

11 H.M. Wiseman and G.J. Milburn, Phys. Rev. Lett. 70, 548

1993; H.M. Wiseman, Phys. Rev. A 49, 2133 1994; 49,

5159E 1994; 50, 4428 1994; H.M. Wiseman and L.K.

Thomsen, Phys. Rev. Lett. 86, 1143 2001.

12 P. Tombesi and D. Vitali, Phys. Rev. A 51, 4913 1995; P.

Goetsch, P. Tombesi, and D. Vitali, ibid. 54, 4519 1996.

FIG. 8. Time evolution of the fidelity and the

parity in the case when the protection scheme isused to filter the one-photon Fock state from

an initial thermal equilibrium state. The stars con-

nected by the dotted line refer to the protected

evolution, while the diamonds linked by the full

line refer to the evolution with no atomic cross-

ing. The cavity relaxation time is Tr1 ms, and

we have considered two velocities v150 m/s

top and v200 m/s bottom. The other param-

eter values are in the text.


052101-11


12/12

13 H. Mabuchi and P. Zoller, Phys. Rev. Lett. 76, 3108 1996.

14 D. Vitali, P. Tombesi, and G.J. Milburn, Phys. Rev. Lett. 79,

2442 1997; Phys. Rev. A 57, 4930 1998.

15 M. Fortunato, J.M. Raimond, P. Tombesi, and D. Vitali, Phys.

Rev. A 60, 1687 1999.

16 U. Haeberlen and J.S. Waugh, Phys. Rev. 175, 453 1968.

17 J.F. Poyatos, J.I. Cirac, and P. Zoller, Phys. Rev. Lett. 77, 4728

1996.

18 N. Lutkenhaus, J.I. Cirac, and P. Zoller, Phys. Rev. A 57, 5481998.

19 A.R.R. Carvalho, P. Milman, R.L. de Matos Filho, and L.

Davidovich, Phys. Rev. Lett. 86, 4988 2001.

20 P. Zanardi, Phys. Rev. A 63, 012301 2001; P. Zanardi and S.

Lloyd, e-print quant-ph/0208132.

21 L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 85, 3520

2000.

22 W.P. Smith, J.E. Reiner, L.A. Orozco, S. Kuhr, and H.M. Wise-

man, Phys. Rev. Lett. 89, 133601 2002.

23 S. Lloyd, Phys. Rev. A 62, 022108 2000.

24 D. Kielpinski et al., Science, 291, 1013 2001.

25 D.G. Cory et al., Phys. Rev. Lett. 81, 2152 1998; E. Knill, R.

Laflamme, R. Martinez, and C. Negrevergne, ibid. 86, 58112001.

26 A.J. Berglund, e-print quant-ph/0010001.

27 L. Viola, E.M. Fortunato, M.A. Pravia, E. Knill, R. Laflamme,

and D.G. Cory, Science, 293, 2059 2001.

28 R.J. Nelson, Y. Weinstein, D. Cory, and S. Lloyd, Phys. Rev.

Lett. 85, 3045 2000.

29 E. Knill, R. Laflamme, and L. Viola, Phys. Rev. Lett. 84, 2525

2000; L. Viola, E. Knill, and R. Laflamme, J. Phys. A 34,

7067 2001.

30 D. Bacon, J. Kempe, D.A. Lidar, and K.B. Whaley, Phys. Rev.

Lett. 85, 1758 2000; D.P. DiVincenzo et al., Nature London

408, 339 2000; J. Kempe, D. Bacon, D.A. Lidar, and K.B.

Whaley, Phys. Rev. A 63, 042307 2001; L.-A. Wu and D.A.

Lidar, Phys. Rev. Lett. 88, 207902 2002; D.A. Lidar and

L.-A. Wu, ibid., 88, 017905 2002.

31 S.L. Braunstein, Phys. Rev. Lett. 80, 4084 1998; S. Lloyd

and J.J. Slotine, ibid. 80, 4088 1998; S.L. Braunstein, Nature

London 394, 47 1998.

32 D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64,

012310 2001.

33 M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A.

Maali, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. 72,

3339 1994.

34 Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J.

Kimble, Phys. Rev. Lett. 75, 4710 1995.

35 M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C.

Wunderlich, J.M. Raimond, and S. Haroche, Phys. Rev. Lett.

77, 4887 1996.

36 X. Maitre, E. Hagley, G. Nogues, C. Wunderlich, P. Goy, M.

Brune, J.M. Raimond, and S. Haroche, Phys. Rev. Lett. 79,

769 1997; P. Bertet, S. Osnaghi, P. Milman, A. Auffeves, P.

Maioli, M. Brune, J.M. Raimond, and S. Haroche, ibid. 88,

143601 2002.

37 B.T.H. Varcoe, S. Brattke, M. Weidinger, and H. Walther, Na-

ture London 403, 743 2000.

38 E. Hagley, X. Maitre, G. Nogues, C. Wunderlich, M. Brune,

J.M. Raimond, and S. Haroche, Phys. Rev. Lett. 79, 1 1997;

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J.M.

Raimond, and S. Haroche, Nature London 400, 239 1999;

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M.


5166 1999; Science, 288, 2024 2000; G. Nogues, A. Raus-

chenbeutel, S. Osnaghi, P. Bertet, M. Brune, J.M. Raimond, S.

Haroche, L.G. Lutterbach, and L. Davidovich, Phys. Rev. A

62, 054101 2000; A. Rauschenbeutel, P. Bertet, S. Osnaghi,

G. Nogues, M. Brune, J.M. Raimond, and S. Haroche, ibid. 64,050301 2001.

39 T. Pellizzari et al., Phys. Rev. Lett. 75, 3788 1995; J.I. Cirac,

P. Zoller, H.J. Kimble, and H. Mabuchi, ibid. 78, 3221 1997;

S.J. van Enk, J.I. Cirac, and P. Zoller, ibid. 78, 4293 1997;

Science, 279, 205 1998.

40 D. F. Walls and G. J. Milburn, Quantum Optics Springer, Ber-

lin, 1994.

41 B.-G. Englert, N. Sterpi, and H. Walther, Opt. Commun. 100,

526 1993.

42 L. Davidovich, M. Brune, J.M. Raimond, and S. Haroche,

Phys. Rev. A 53, 1295 1996.

43 L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547

1997; Opt. Express 3, 147 1998.

44 P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M.


200402 2002.

45 N. F. Ramsey, Molecular Beams Oxford University Press,

New York, 1985.

46 J.M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73,

565 2001.

47 S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M.

Raimond, and S. Haroche, Phys. Rev. Lett. 87, 037902 2001.

48 D.F. Walls and G.J. Milburn, Phys. Rev. A 31, 2403 1985.

49 P. Domokos, M. Brune, J.M. Raimond, and S. Haroche, Eur.

Phys. J. D 1, 1 1998.


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